Peter Easthope

Hi,

In Mod-01, Lec-02, beginning at 54:10 Balakrishnan discusses the 
"inverted oscillator" having energy 1/2 m qdot^2 - 1/2 m omega^2 q^2.
 
https://www.youtube.com/watch?v=8X1x9RLaaxc

Discussed further in Lec-03.

Is the "inverted oscillator" only fictitious?  If a realization 
exists, I must have a mental block to imagine it.

Thx,                           ... P.
 

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... 'space' alone does not admit of translation.
Something translates or is translated in or through space (or rotates in it).

An instance or example: do an experiment in London, Ontario and the same experiment in London, England.  Assuming conditions are genuinely duplicated (or that bias is avoided) the results in the two cases will be the same.

8 hours ago, studiot said:

Did you get your reference to Noether from somewhere like this ?

 Didn't find the source for that but it might be under copyright.  This is readily available.  https://en.wikipedia.org/wiki/Noether's_theorem

Regards,      ... P.

I need to study the Lagrangian calculation more. 

Unfortunately the engineering curriculum focuses on "practical" calculations at the expense of the abstract formulations.

Thanks for the help,         ... P.

1 hour ago, swansont said:

Is the mass varying with position in the first example?

Is the mass varying with time in the second example?

I imagined constancy of mass or masses

1 hour ago, swansont said:

You wouldn’t be applying symmetry to these problems if mass were varying, as their Lagrangians are not constant, i.e. not symmetric under the relevant translation.

OK, thanks.  I need to understand the Lagrangian calculation better,       ... P.

20 hours ago, studiot said:

So tell us what expression of the theorem you have as it is difficult to offer generalisations if the question is too broad.

A convenient elementary example: a Newtonian CO2 puck on a level table.  Non-zero invariant mass, m. 

(S) Verification of time invariance: verify that the path is a straight line; verify that velocity is constant.  Assume m is constant or ignore it?

(C) Conservation of energy: assuming the table is level, gravitational potential is constant.  Evaluate (m v^2)/2 over a time sequence and verify constancy.  Assume m is constant?  Otherwise, how is m measured during motion?

The invariant mass is assumed to be constant in S?  Also in C?

Thx,          ... P.

Make that "air puck" rather than "CO2 puck",       ... P.

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How can Noether's Theorem be reconciled with dimensional analysis?

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Why does it need to be?

Only an interest.  Not a necessity.

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Photons disagree with you.

Good point.  Several details to understand.  More below.
 

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The theorem of conservation of energy is only one component, the time

component, of a law which is invariant for Lorenz trnasformations, the
other components being the space components which express the
conservation of momentum.

The total energy as well as the total momentum remains unchanged;
 ......

The symmetry of the four dimensional energy momentum tensor tells us
the the momentum density = 1/c2 times the energy flux.

 

Thanks.  Definitely helpful.

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But your question may have nothing to do with the symmetries (and
invariances) of this particular transformation.

Yes, my understanding is meagre.

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So please provide the context in which you have asked this question.

I don't have a specific context.  Just trying to understand the theorem.

The conservations, C, involve spacetime and mass whereas the
symmetries, S, involve spacetime but not mass.

Equivalence requires two implications: C => S and S => C.

C => S entails removal of mass.
S => C entails introduction of mass.

The presentations I've seen avoid attention to mass. I'm not
claiming that's incorrect; only that I'm interested to understand
the involvement of mass better.

Thanks for the feedback,                    ... P.

Hi,

How can Noether's Theorem be reconciled with dimensional analysis?

For example, invariance under spatial translation is equivalent to
conservation of momentum.  Spacial translation has no involvement of
mass but mass is essential to momentum.  

How can the invariance independent of mass be equivalent to
conservation of a quantity where mass is essential?

Thanks,                 ... Peter E.